How Interim Information Fraction Changes Predictive Success

Why Information Fraction Matters

An interim result is observed before all planned information has accumulated. The question “How likely is the trial to succeed?” therefore depends not only on the current data, but also on how much information remains to change the final result. With substantial information still to come, a favorable or unfavorable interim estimate may move considerably. Near the planned end, the final result is more constrained by what has already been observed.

This tutorial studies that relationship while holding the final success rule fixed. The examples use a normal model to make the role of information fraction explicit. The calculations describe uncertainty about the eventual decision under the Bayesian model; they do not say that the treatment effect itself changes as the trial proceeds.

Definition: The interim information fraction $f$ is the information accumulated at an interim analysis divided by the information planned at the final analysis. In a fixed-variance normal model with information $I_t$ at interim and $I_F$ at the planned final analysis, $f=I_t/I_F$, where $0<f<1$. It is not necessarily the proportion of participants enrolled: information per participant can vary, for example, with outcome variability, follow-up, or event accrual.

A Normal-Model Result

Suppose the parameter of interest is a treatment effect $\theta$, with positive values representing benefit. Assume a normal likelihood with known variance and independent increments of information: the data accumulated after the interim analysis provide information $I_F-I_t$, independent of the interim data conditional on $\theta$. Use a flat prior for $\theta$, so that after interim data $D_t$, the posterior is $N(m_t,1/I_t)$. The final posterior variance is then $1/I_F$. These assumptions are a useful model for deriving the relationship; they are not automatic for every clinical endpoint or trial design.

Let the fixed final success rule be that the final posterior probability of benefit exceeds $q$, where $q$ is fixed in advance and $0.5<q<1$. Write $c=\Phi^{-1}(q)$, where $\Phi$ is the standard normal cumulative distribution function. Because the final posterior standard deviation is $1/\sqrt{I_F}$, final success is equivalent to the final posterior mean exceeding $c/\sqrt{I_F}$.

P(\theta>0\mid D_F)>q \quad\Longleftrightarrow\quad m_F>\frac{c}{\sqrt{I_F}}.

Here $D_F$ denotes the eventual full data and $m_F$ their final posterior mean. Predictive success at interim is the probability, conditional on the data already observed, that this fixed final rule will be met after the remaining data are observed. It averages over the posterior uncertainty about $\theta$ and the distribution of future observations.

Result: Under the stated normal model, the predictive distribution of the final posterior mean is $m_F\mid D_t\sim N(m_t,1/I_t-1/I_F)$. Thus, for $0<f<1$, predictive success is $$ P(\mathrm{final\ success}\mid D_t) = \Phi\left( \frac{m_t-c/\sqrt{I_F}}{\sqrt{1/I_t-1/I_F}} \right) = \Phi\left( \frac{\sqrt{I_F}m_t-c}{\sqrt{1/f-1}} \right). $$

The denominator is the predictive standard deviation of the final posterior mean, not the final posterior standard deviation. It decreases as $f$ increases: less information remains to move the final mean. The expression also shows that information fraction alone does not determine predictive success. The current estimate $m_t$, the planned final information, and the fixed criterion $q$ all matter.

Why the Formula Holds

Given the interim posterior, the future sufficient statistic $Y_R$ has likelihood $Y_R\mid\theta\sim N(\theta,1/(I_F-I_t))$. Combining that likelihood with the interim posterior gives the final posterior mean $m_F=\{I_t m_t+(I_F-I_t)Y_R\}/I_F$. Under the interim posterior predictive distribution, $Y_R\mid D_t$ is normal with mean $m_t$ and variance $1/I_t+1/(I_F-I_t)$.

Taking the mean of the expression for $m_F$ gives $m_t$. Its variance follows by multiplying the variance of $Y_R$ by the square of its coefficient:

\begin{aligned} \operatorname{Var}(m_F\mid D_t) &=\left(\frac{I_F-I_t}{I_F}\right)^2 \left(\frac{1}{I_t}+\frac{1}{I_F-I_t}\right)\\ &=\frac{1}{I_t}-\frac{1}{I_F}. \end{aligned}

The second equality follows by combining the terms in parentheses to obtain $I_F/\{I_t(I_F-I_t)\}$. Since $m_F$ is normally distributed and success is the event $m_F>c/\sqrt{I_F}$, standardizing that boundary gives the displayed predictive-success formula. This derivation also clarifies why the final posterior criterion must be held fixed when comparing interim analyses.

Worked Example: An Early Interim Analysis

Consider a hypothetical normal-outcome trial with planned information $I_F=100$. At an early interim, $I_t=25$, so $f=25/100=0.25$. Suppose the current posterior is $N(0.1000,1/25)$, and the fixed final rule is $P(\theta>0\mid D_F)>0.95$. Thus $c=\Phi^{-1}(0.95)\approx1.6449$.

The final posterior mean must exceed $1.6449/\sqrt{100}=0.1645$. The predictive standard deviation of that mean is $\sqrt{1/25-1/100}=\sqrt{0.03000}=0.1732$. Therefore:

P(\mathrm{final\ success}\mid D_t) = \Phi\left(\frac{0.1000-0.1645}{0.1732}\right) = \Phi(-0.372)\approx0.355.

Under the model and the data observed so far, the chance of meeting the specified final posterior rule is about 35.5%. The current posterior probability of benefit is $\Phi(0.1000/0.2000)=\Phi(0.5000)\approx0.6915$, which is favorable relative to 0.5 but does not itself answer whether the final rule is likely to be met. The early analysis leaves substantial room for future data to move the final estimate in either direction.

Worked Example: A Late Interim Analysis With the Same Current Mean

Keep the planned information $I_F=100$, the final 0.95 posterior criterion, and the current posterior mean $m_t=0.1000$, but consider a later analysis at $I_t=80$, so $f=0.80$. The current posterior variance is $1/80=0.01250$. The final boundary remains $0.1645$, but the predictive standard deviation is now $\sqrt{1/80-1/100}=\sqrt{0.002500}=0.05000$.

P(\mathrm{final\ success}\mid D_t) = \Phi\left(\frac{0.1000-0.1645}{0.05000}\right) = \Phi(-1.290)\approx0.0985.

The predictive probability is about 9.85%, lower than at the early analysis even though the interim posterior mean is the same. With less information remaining, the final posterior mean is less likely to move above the unchanged boundary. The current posterior probability of benefit is now $\Phi(0.1000/\sqrt{0.01250})=\Phi(0.8944)\approx0.8145$. That current probability is higher than in the early example because the same mean is estimated more precisely, while predictive success is lower because the final boundary remains above the mean and there is less opportunity for the mean to cross it.

Separating Current Evidence From Predictive Success

A current posterior probability concerns the parameter given the data already observed. Predictive success concerns a future decision: the probability that the final data will produce a posterior meeting a specified rule. They use related information, but answer different questions. A high current probability of benefit can coexist with low predictive success if the final criterion is demanding and little information remains. Conversely, an uncertain early estimate can have a nontrivial probability of eventual success because the remaining data could substantially shift the final posterior.

Worked Example: Two Interim Quantities and a Fixed Final Boundary

For a separate hypothetical illustration, suppose the final success rule is a fixed posterior-mean boundary of $0.1839$. Equivalently, with a fixed final posterior standard deviation of $0.1327$, it is a fixed posterior-tail rule of approximately $P(\theta>0\mid D_F)>0.917$. Consider an interim posterior variance of $0.02000$; the corresponding predictive standard deviation of the final posterior mean is $0.04880$, since the remaining information gives final variance $0.02000-0.04880^2=0.01762$ (rounding).

In one interim state, let the current posterior be $N(0.1600,0.02000)$. The current posterior probability of benefit is:

P(\theta>0\mid D_t) = \Phi\left(\frac{0.1600}{\sqrt{0.02000}}\right) = \Phi(1.1314)\approx0.8711.

In a second interim state with the same posterior variance and remaining information, let the current posterior mean instead be $0.2000$. Its predictive success probability is:

P(\mathrm{final\ success}\mid D_t) = 1-\Phi\left(\frac{0.1839-0.2000}{0.04880}\right) = 1-\Phi(-0.3299) \approx0.6293.

The current evidence calculation and predictive calculation refer to distinct quantities; the latter depends on the fixed final boundary and uncertainty in the future final mean. These figures illustrate how to calculate each quantity, not a universal mapping from one to the other. In an actual analysis, use the model’s unrounded variances and boundary throughout, then round only the reported results.

Reading the Information-Fraction Pattern

For the normal model, if the information fraction $f$, planned information, current mean, and final rule are specified, the formula gives the predictive probability directly. The practical pattern depends on where the current mean lies relative to the final boundary:

Current mean below the boundary: predictive success is below 0.5. As information accumulates while the mean remains below the boundary, the opportunity for future movement across it generally contracts.
Current mean above the boundary: predictive success is above 0.5. As information accumulates, the final result becomes more likely to remain on that side, provided the current mean itself does not change materially.
Current mean near the boundary: predictive success can be near 0.5, and small changes in the estimate or the fixed criterion can noticeably change the probability.

These are conditional patterns, not a claim that predictive success must move monotonically across actual interim looks. The observed $m_t$ changes as data arrive. A sufficiently large change in the interim estimate can outweigh the shrinking predictive standard deviation; therefore a trial’s calculated predictive success may rise or fall from one analysis to the next.

Assumptions, Diagnostics, and Alternatives

The calculation relies on a specified likelihood and prior, a fixed planned final information level, and a future-data model consistent with the information increments. The simple expression above uses a flat prior and known-variance normal observations. With a proper normal prior or a different endpoint model, the predictive distribution must be derived from that model; one should not substitute the participant fraction for $f$ without justification. Missing outcomes, changes in measurement variability, unplanned changes to the estimand, or outcome-dependent stopping can also make the simple information accounting inadequate.

Before interpreting a probability, check that interim and final information use the same endpoint and analysis scale, that the remaining information is positive, and that the boundary corresponds exactly to the planned final rule. In the normal model, a useful numerical check is that the predictive variance $1/I_t-1/I_F$ is positive for $I_t<I_F$, approaches zero as $f$ approaches one, and is larger at an earlier look. A negative variance or a predictive probability outside zero to one signals a calculation or implementation error. Also compare the assumed variance with blinded or otherwise appropriate design information; a substantial mismatch can make the nominal fraction misleading.

The principal alternative is to inspect a current posterior probability or apply a current-posterior go/no-go rule. That is appropriate when the decision is explicitly about current evidence, but it does not average over data still to come. A predictive rule instead asks whether the final criterion is likely to be met and can support a continue-or-stop decision. Either approach must be tied to the trial’s decision costs and prespecified policy; neither probability alone establishes clinical importance or guarantees future success. The normal formula is also a model-based calculation, not an operating-characteristic evaluation across hypothetical trials.

Common pitfall: “The current posterior probability of benefit is 0.81, so the probability of meeting the final success rule is 0.81.” This treats two different events as if they were the same. The current probability is $P(\theta>0\mid D_t)$; predictive success is the probability that the future final posterior will satisfy its specified rule. Calculate the latter by integrating over future outcomes under the predictive distribution.

Check Your Understanding

Try each question first, then compare your reasoning with the solutions.

In the normal model, planned information is 120 and interim information is 30. What is the information fraction, and why need it not equal the enrolled-participant fraction?
At an early analysis, $I_t=20$, $I_F=100$, and $m_t=0.15$. For a final 0.95 posterior criterion for $\theta>0$, calculate the final boundary and predictive success.
With the same plan and final criterion, suppose a later analysis has $I_t=80$ and $m_t=0.15$. Calculate predictive success and explain why it differs from Question 2.
Does predictive success have to increase at each interim analysis? Explain what can cause it to rise or fall.
A colleague interprets predictive success of 0.30 as “there is a 30% posterior probability that the treatment works.” Correct this interpretation.

Solutions. (1) $f=30/120=0.25$. Participant counts need not be proportional to information when information per participant varies. (2) The boundary is $1.6449/\sqrt{100}=0.1645$. The predictive standard deviation is $\sqrt{1/20-1/100}=\sqrt{0.04000}=0.2000$, so success is $\Phi((0.1500-0.1645)/0.2000)=\Phi(-0.0725)\approx0.4711$. (3) The predictive standard deviation is $\sqrt{1/80-1/100}=0.05000$, giving $\Phi((0.1500-0.1645)/0.05000)=\Phi(-0.290)\approx0.3859$. Less information remains, so the final mean is less variable and less able to cross the boundary; the interim estimate is unchanged. (4) No. The predictive distribution narrows as information accumulates, but the observed interim estimate can move in either direction, changing its distance from the boundary. (5) A value of 0.30 is the conditional probability, under the specified model and current data, that the eventual final posterior will meet the fixed success rule. It is not the posterior probability that the treatment works.